Class 8 RD Sharma Solutions Chapter 4 Cubes and Cube Roots  Exercise 4.2 (original) (raw)

Last Updated : 12 Sep, 2024

Exercise 4.2 of Chapter 4 in RD Sharma's Class 8 Mathematics textbook delves into the concepts of cubes and cube roots, offering students a comprehensive exploration of these fundamental mathematical ideas. This exercise is designed to enhance students' understanding of perfect cubes, calculation of cube roots, and application of these concepts in various problem-solving scenarios. Through a series of carefully crafted questions, students will develop their skills in identifying perfect cubes, computing cube roots of whole numbers, fractions, and decimals, and solving real-world problems that involve cubic measurements and relationships. The exercise aims to build a strong foundation in this area of mathematics, preparing students for more advanced concepts in higher grades.

**Question 1. Find the cubes of:

****(i) -11**
****(ii) -12**
****(iii) -21**

**Solution:

**i) Cube of -11 = (-11)3

= -11 × -11 × -11 = -1331

**ii) Cube of -12 = (-12)3

= -12 × -12 ×-12 = -1728

**iii) Cube of -21 = (-21)3

= -21 × -21 ×-21 = -9261

**Question 2. Which of the following numbers are cubes of negative integers.

****(i) -64**
****(ii) -1056**
****(iii) -2197**
****(iv) -2744**
****(v) -42875**

**Solution:

In order to find out there the given negative number is a perfect cube or not, we need to check if its corresponding positive number is a perfect cube.

**i) -64

Let's first check whether 64 is a perfect cube or not.

Prime factorization of 64

64 = 2 × 2 × 2 × 2 × 2 × 2

Also, 64 = (2 × 2 × 2) × (2 × 2 × 2)

Since, 64 can be completely grouped in triplets of the equal factors,

So, 64 is a perfect cube of 4.

**Hence, -64 is a perfect cube of negative number i.e -4.

**ii) -1056

Let's first check whether 1056 is a perfect cube or not.

Prime factorization of 1056

1056 = 2 × 2 × 2 × 2 × 2 × 3 × 11

Also, 1056 = (2 × 2 × 2) × 2 × 2 × 3 × 11

Since, 1056 can't be completely grouped in triplets of the equal factors,

So, 1056 is not a perfect cube.

**Hence, -1056 is a not perfect cube of a negative number.

**iii) -2197

Let's first check whether 2197 is a perfect cube or not.

Prime factorization of 2197

2197 = 13 × 13 × 13

Also, 2197 = (13 × 13 × 13)

Since, 2197 can be completely grouped in triplets of the equal factors,

So, 2197 is a perfect cube of 13.

**Hence, -2197 is a perfect cube of negative number i.e -13.

**iv) -2744

Let's first check whether 2744 is a perfect cube or not.

Prime factorization of 2744

2744 = 2 × 2 × 2 × 7 × 7 × 7

Also, 2744 = (2 × 2 × 2) × (7 × 7 × 7)

Since, 2744 can be completely grouped in triplets of the equal factors,

So, 2744 is a perfect cube of 14.

**Hence, -2744 is a perfect cube of negative number i.e -14.

**v) -42875

Let's first check whether 42875 is a perfect cube or not.

Prime factorization of 42875

42875 = 5 × 5 × 5 × 7 × 7 × 7

Also, 42875 = (5 × 5 × 5) × (7 × 7 × 7)

Since, 42875 can be completely grouped in triplets of the equal factors,

So, 42875 is a perfect cube of 35.

**Hence, -42875 is a perfect cube of negative number i.e -35.

**Question 3. Show that the following integers are cubes of negative integers. Also, find the integer whose cube is the given integer :

****(i) -5832 (ii) -2744000**

**Solution:

**i) -5832

Let's first check whether 5832 is a perfect cube or not.

Prime factorization of 5832

5832 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3

Also, 5832 = (2 × 2 × 2) × (3 × 3 × 3) × (3 × 3 × 3)

Since, 5832 can be completely grouped in triplets of the equal factors,

So, 5832 is a perfect cube of 18.

**Hence, -5832 is a perfect cube of negative number i.e -18.

**ii) 2744000

Let's first check whether 2744000 is a perfect cube or not.

Prime factorization of 2744000

2744000 = 2 × 2 × 2 × 7 × 7 × 7 × 2 × 2 × 2 × 5 × 5 × 5

Also, 2744000 = (2 × 2 × 2) × (7 × 7 × 7) × (2 × 2 × 2) × (5 × 5 × 5)

Since, 2744000 can be completely grouped in triplets of the equal factors,

So, 2744000 is a perfect cube of 140.

**Hence, -2744000 is a perfect cube of negative number i.e -140.

**Question 4. Find the cube of :

****(i) 7/9**
****(ii) -8/11**
****(iii) 12/7**
****(iv) -13/8**
****(v) 12/5**
****(vi) 13/4**
****(vii) 0.3**
****(vii) 1/5**
****(ix) 0.08**
****(x) 2.1**

**Solution:

**i) Cube of 7/9 will be 7/9 × 7/9 × 7/9

= 343/729

**Hence, the cube of 7/9 is 343/729

**ii) Cube of -8/11 will be -8/11 × -8/11 × -8/11

= -512/1331

**Hence, the cube of -8/11 is -512/1331

**iii) Cube of 12/7 will be 12/7 × 12/7 × 12/7

= 1728/343

**Hence, the cube of 12/7 is 1728/343

**iv) Cube of -13/8 will be -13/8 × -13/8 × -13/8

= -2197/512

**Hence, the cube of -13/8 is -2197/512

**v) Cube of 12/5 will be 12/5 × 12/5 × 12/5

= 1728/125

**Hence, the cube of 12/5 is 1728/125

**vi) Cube of 13/4 will be 13/4 × 13/4 × 13/4

= 2197/64

**Hence, the cube of 13/4 is 2197/64

**vii) 0.3 = 3/10

So, Cube of 3/10 will be 3/10 × 3/10 × 3/10

= 27/1000 = 0.027

**Hence, the cube of 0.3 is 0.027

**viii) 1.5 = 15/10

So, Cube of 15/10 will be 15/10 × 15/10 × 15/10

= 3375/1000 = 3.375

**Hence, the cube of 1.5 is 3.375

**ix) 0.08 = 8/100

So, Cube of 8/100 will be 8/100 × 8/100 × 8/100

= 512/1000000 = 0.000512

**Hence, the cube of 0.08 is 0.000512

**x) 2.1 = 21/10

So, Cube of 21/10 will be 21/10 × 21/10 × 21/10

= 9261/1000 = 9.261

**Hence, the cube of 2.1 is 9.261

**Question 5. Which of the following numbers are cubes of rational numbers :

****(i) 27/64**
****(ii) 125/128**
****(iii) 0.001331**
****(iv) 0.04**

**Solution:

**i) 27/64

Factorization of 27/64 will be

27/64 = (3 × 3 × 3)/(4 × 4 × 4) = (3/4)3

It means the cube of 3/4 is 27/64

**Hence, we can say that 27/64 is a cube of rational number i.e 3/4

**ii) 125/128

Factorization of 125/128 will be

125/128 = (5 × 5 × 5)/(2 × 2 × 2) × (2 × 2 × 2) × 2 = 53/23 × 23 × 2

It means 125/128 is not perfect cube

**Hence, we can say that 125/158 is not a cube of rational number

**iii) **0.001331

0.001331 = 1331/1000000

Factorization of 1331/1000000 will be

1331/1000000 = (11 × 11 × 11)/(10 × 10 × 10) × (10 × 10 × 10) =113/103 × 103

It means the cube of 11/100 is 1331/1000000

**Hence, we can say that 0.001331 is a cube of rational number i.e 0.11

**iv) 0.04

0.04 = 4/100

Factorization of 4/100 = 2 × 2/ 10 × 10

It means 4/100 is not a perfect cube

**Hence, we can say that 0.04 is not a cube of a rational number

Summary

Exercise 4.2 provides a thorough practice session on cubes and cube roots, covering a wide range of applications and problem types. Students engage with tasks such as calculating cube roots of perfect cubes, identifying numbers that are perfect cubes, and solving word problems that require the application of cube and cube root concepts. The exercise also introduces more complex scenarios, including working with decimal and fractional cube roots, understanding the relationship between a cube's volume and its side length, and applying these concepts to real-world situations. By completing this exercise, students not only enhance their computational skills but also develop a deeper conceptual understanding of cubes and cube roots. This knowledge forms a crucial foundation for future mathematical studies, particularly in areas such as algebra, geometry, and higher-level problem-solving.